Calculation of Output Noise Variances for Discrete Time-Invariant Filters

Certain calculations to minimize output noise variance are introduced. Many applied problems in sampled data systems require that data be smoothed in the presence of noise for the prediction of future positions, velocities, or accelerations. Smoothing coefficients in discrete time-invariant filters are computed to minimize the output noise variance, but under the constraints that the function and derivatives be predicted ahead. The output noise variance is seen to be a function of the input noise, the number of input signals (N+1) that the filter has to smooth, and the prediction time αT. Four examples are given in the derivation of smoothing coefficients for step and ramp inputs subjected to either almost white noise or Gaussian-Markoff noise. The examples illustrate the number of constraint relations that the filter smoothing coefficients must satisfy for function and/or derivative convergence under noise-free conditions. The smoothing coefficients are also a function of the type of noise input into the system or the discrete filter. From the examples, it can be observed that as N becomes larger, the output noise variance becomes smaller, but the computation time is increased.